Twisted Tensors of Hopf Algebras (24rit600)



Amrei Oswald (University of Washington)


The Banff International Research Station will host the "Twisted Tensors of Hopf Algebras" workshop in Banff from March 17 to March 30, 2024.

Our collaboration concerns the study of the representation theory of Hopf algebras via twisted tensor products. It began during the 20th International Conference on Representations of Algebras. During the commute from Montevideo to Buenos Aires we talked about how the COVID pandemic had affected our workflow and our research collaborations, how it affected our ability to give physical talks when on the job market, and how Zoom burnout and lack of a proper response to these issues at the University level hindered our productivity. Given that we both worked in quantum symmetries, we thought it may be interesting to tackle questions that piqued our common interests.

After some conversations, we decided that understanding quantum groups as twisted tensor products was a first reasonable stepping stone towards larger and more complex projects. Twisted tensor products are a noncommutative analogue of the usual tensor product, and encapsulate the algebra structure of a huge variety of interesting examples. This has already been proven useful to compute the Hochschild cohomology of algebras such as the Jordan plane, as well as to understand quantum complete intersections, just to name two applications. However, in general, twisted tensor products do not capture information about the Hopf algebra structure. Our goal was to establish a Hopf algebra structure on twisted tensor products, and use this to obtain information about a given algebra by decomposing it into simpler components. A motivating endpoint was to decompose the Balmer spectrum of the stable module category of a quantum group in terms of its components as a twisted tensor product.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.